On the 4-permutational property (Q1091466)

The finite group lies in the class \(P_ 4\) if, given elements \(x_ 1,...,x_ 4\) of G, there is a permutation \(\sigma\) of \(\{\) 1,2,3,4\(\}\) such that \(x_{1\sigma}x_{2\sigma}x_{3\sigma}x_{4\sigma}=x_ 1x_ 2x_ 3x_ 4\). Using the N-group paper, it is shown that all such groups are soluble. Further, it is shown by less regal means that for such a group G, the exponent of the Sylow 2-subgroups of G/F(G) is a divisor of 4, G' is nilpotent, and G/F(G) is an Abelian \(\{\) 2,3\(\}\)- group. No motivation for or application of these results is given.